1. Copyright © Cengage Learning. All rights reserved. 11 Multifactor Analysis of Variance

2. Copyright © Cengage Learning. All rights reserved. 11.4 2 p Factorial Experiments

3. 3 If an experimenter wishes to study simultaneously the effect of p different factors on a response variable and the factors have I 1, I 2,…, I p levels, respectively, then a complete experiment requires at least I 1  I 2,  …  I p observations. In such situations, the experimenter can often perform a “screening experiment” with each factor at only two levels to obtain preliminary information about factor effects. An experiment in which there are p factors, each at two levels, is referred to as a 2 p factorial experiment.

4. 4 2 3 Experiments

5. 5 Let X ijkl and x ijkl refer to the observation from the l th replication, with factors A, B, and C at levels i, j, and k, respectively. The model for this situation is for i = 1, 2; j = 1, 2; k = 1, 2; l = 1, …, n. The  ijkl ’s are assumed independent, normally distributed, with mean 0 and variance  2. (11.14)

6. 6 2 3 Experiments Because there are only two levels of each factor, the side conditions on the parameters of (11.14) that uniquely specify the model are simply stated: and the like. These conditions imply that there is only one functionally independent parameter of each type (for each main effect and interaction).

7. 7 2 3 Experiments For example,  2 = –  1, whereas Because of this, each sum of squares in the analysis will have 1 df. The parameters of the model can be estimated by taking averages over various subscripts of the X ijkl ’s and then forming appropriate linear combinations of the averages.

8. 8 2 3 Experiments For example, and

9. 9 2 3 Experiments Each estimator is, except for the factor 1/(8n), a linear function of the cell totals (X ijk ’s) in which each coefficient is +1 or –1, with an equal number of each; such functions are called contrasts in the X ijk ’s. Furthermore, the estimators satisfy the same side conditions satisfied by the parameters themselves. For example,

10. 10 Example 12 In an experiment to investigate the compressive strength properties of cement–soil mixtures, two different aging periods were used in combination with two different temperatures and two different soils. Two replications were made for each combination of levels of the three factors, resulting in the following data:

11. 11 Example 12 The computed cell totals are x 111 = 884, x 211 = 1349, x 121 = 1037, x 221 = 1501, x 112 = 819, x 212 = 1475, x 122 = 1123 and x 222 = 1547, so x…. = 9735. Then = (884 – 1349 + 1037 – 1501 + 819 – 1475 + 1123 – 1547)/16 = –125.5625 = cont’d

12. 12 Example 12 = (884 – 1349 – 1037 + 1501 + 819 – 1475 – 1123 – 1547)/16 = –14.5625 = The other parameter estimates can be computed in the same manner. cont’d

13. 13 2 3 Experiments Analysis of a 2 3 Experiment Sums of squares for the various effects are easily obtained from the parameter estimates. For example, and

14. 14 2 3 Experiments Since each estimate is a contrast in the cell totals multiplied by 1/(8n), each sum of squares has the form (contrast) 2 /(8n). Thus to compute the various sums of squares, we need to know the coefficients (+1 or –1) of the appropriate contrasts.

15. 15 2 3 Experiments The signs ( + or –) on each X ijk in each effect contrast are most conveniently displayed in a table. We will use the notation (1) for the experimental condition i = 1, j = 1, k = 1, a for i = 2, j = 1, k = 1, ab for i = 2, j = 2, k = 1, and so on.

16. 16 2 3 Experiments If level 1 is thought of as “low” and level 2 as “high,” any letter that appears denotes a high level of the associated factor. Each column in Table 11.10 gives the signs for a particular effect contrast in the x ijk ’s associated with the different experimental conditions. Signs for Computing Effect Contrasts Table 11.10

17. 17 2 3 Experiments In each of the first three columns, the sign is + if the corresponding factor is at the high level and – if it is at the low level. Every sign in the AB column is then the “product” of the signs in the A and B columns, with (+)(+) = (–)(–) = + and (+)(–) = (–)(+) = –, and similarly for the AC and BC columns. Finally, the signs in the ABC column are the products of AB with C (or B with AC or A with BC).

18. 18 2 3 Experiments Thus, for example, AC contrast = + X 111 – X 211 + X 121 – X 221 – X 112 + X 212 – X 122 + X 222 Once the seven effect contrasts are computed, Software for doing the calculations required to analyze data from factorial experiments is widely available (e.g., Minitab).

19. 19 2 3 Experiments Alternatively, here is an efficient method for hand computation due to Yates. Write in a column the eight cell totals in the standard order, as given in the table of signs, and establish three additional columns. In each of these three columns, the first four entries are the sums of entries 1 and 2, 3 and 4, 5 and 6, and 7 and 8 of the previous columns. The last four entries are the differences between entries 2 and 1, 4 and 3, 6 and 5, and 8 and 7 of the previous column.

20. 20 2 3 Experiments The last column then contains x…. and the seven effect contrasts in standard order. Squaring each contrast and dividing by 8n then gives the seven sums of squares.

21. 21 Example 13 Example 12 continued… Since n = 2, 8n = 16. Yates’s method is illustrated in Table 11.11. Yates’s Method of Computation Table 11.11

22. 22 Example 13 From the original data, and so SST = 6,232,289 – 5,923,139.06 = 309,149.94 SSE = SST – [SSA + … + SSABC] = 309,149.94 – 292,036.42 = 17,113.52 cont’d

23. 23 Example 13 The ANOVA calculations are summarized in Table 11.12. ANOVA Table for Example 13 Table 11.12 cont’d

24. 24 Example 13 Figure 11.10 shows SAS output for this example. Only the P-values for age (A) and temperature (B) are less than.01, so only these effects are judged significant. SAS output for strength data for Example 13 Figure 11.10 cont’d

25. 25 2 p Experiments for p > 3

26. 26 2 p Experiments for p > 3 The analysis of data from a 2 p experiment with p > 3 parallels that of the three-factor case. For example, if there are four factors A, B, C, and D, there are 16 different experimental conditions. The first 8 in standard order are exactly those already listed for a three-factor experiment. The second 8 are obtained by placing the letter d beside each condition in the first group.

27. 27 2 p Experiments for p > 3 Yates’s method is then initiated by computing totals across replications, listing these totals in standard order, and proceeding as before; with p factors, the pth column to the right of the treatment totals will give the effect contrasts. For p > 3, there will often be no replications of the experiment (so only one complete replicate is available). One possible way to test hypotheses is to assume that certain higher-order effects are absent and then add the corresponding sums of squares to obtain an SSE.

28. 28 2 p Experiments for p > 3 Such an assumption can, however, be misleading in the absence of prior knowledge (see the book by Montgomery listed in the chapter bibliography). An alternative approach involves working directly with the effect contrasts. Each contrast has a normal distribution with the same variance. When a particular effect is absent, the expected value of the corresponding contrast is 0, but this is not so when the effect is present.

29. 29 2 p Experiments for p > 3 The suggested method of analysis is to construct a normal probability plot of the effect contrasts (or, equivalently, the effect parameter estimates, since estimate = contrast/2 p when n = 1). Points corresponding to absent effects will tend to fall close to a straight line, whereas points associated with substantial effects will typically be far from this line.

30. 30 Example 14 The accompanying data is from the article “Quick and Easy Analysis of Unreplicated Factorials” (Technometrics, 1989: 469–473). The four factors are A = acid strength, B = time, C = amount of acid, and D = temperature, and the response variable is the yield of isatin. The observations, in standard order, are.08,.04,.53,.43,.31,.09,.12,.36,.79,.68,.73,.08,.77,.38,.49, and.23.

31. 31 Example 14 Table 11.13 displays the effect estimates as given in the article (which uses contrast/8 rather than contrast/16). Effect Estimates for Example 14 Figure 11.13 cont’d

32. 32 Example 14 Figure 11.11 is a normal probability plot of the effect estimates. All points in the plot fall close to the same straight line, suggesting the complete absence of any effects (we will shortly give an example in which this is not the case). A normal probability plot of effect estimates from Example 11.14 Figure 11.11 cont’d

33. 33 2 p Experiments for p > 3 Visual judgments of deviation from straightness in a normal probability plot are rather subjective. The article cited in Example 14 describes a more objective technique for identifying significant effects in an unreplicated experiment.

34. 34 Confounding

35. 35 Confounding It is often not possible to carry out all 2 p experimental conditions of a 2 p factorial experiment in a homogeneous experimental environment. In such situations, it may be possible to separate the experimental conditions into 2 r homogeneous blocks (r < p), so that there are 2 p – r experimental conditions in each block. The blocks may, for example, correspond to different laboratories, different time periods, or different operators or work crews.

36. 36 Confounding In the simplest case, p = 3 and r = 1, so that there are two blocks, with each block consisting of four of the eight experimental conditions. As always, blocking is effective in reducing variation associated with extraneous sources. However, when the 2 p experimental conditions are placed in 2 r blocks, the price paid for this blocking is that 2 r – 1 of the factor effects cannot be estimated. This is because 2 r – 1 factor effects (main effects and/or interactions) are mixed up, or confounded, with the block effects.

37. 37 Confounding The allocation of experimental conditions to blocks is then usually done so that only higher-level interactions are confounded, whereas main effects and low-order interactions remain estimable and hypotheses can be tested. To see how allocation to blocks is accomplished, consider first a 2 3 experiment with two blocks (r = 1) and four treatments per block. Suppose we select ABC as the effect to be confounded with blocks.

38. 38 Confounding Then any experimental condition having an odd number of letters in common with ABC, such as b (one letter) or abc (three letters), is placed in one block, whereas any condition having an even number of letters in common with ABC (where 0 is even) goes in the other block. Figure 11.12 shows this allocation of treatments to the two blocks. Confounding ABC in a 2 3 experiment Figure 11.12

39. 39 Confounding In the absence of replications, the data from such an experiment would usually be analyzed by assuming that there were no two-factor interactions (additivity) and using SSE = SSAB + SSAC + SSBC with 3 df to test for the presence of main effects. Alternatively, a normal probability plot of effect contrasts or effect parameter estimates could be examined. Most frequently, though, there are replications when just three factors are being studied. Suppose there are u replicates, resulting in a total of 2 r  u blocks in the experiment.

40. 40 Confounding Then after subtracting from SST all sums of squares associated with effects not confounded with blocks (computed using Yates’s method), the block sum of squares is computed using the 2 r  u block totals and then subtracted to yield SSE (so there are 2 r  u – 1 df for blocks).

41. 41 Example 15 The article “Factorial Experiments in Pilot Plant Studies” (Industrial and Eng. Chemistry, 1951: 1300–1306) reports the results of an experiment to assess the effects of reactor temperature (A), gas throughput (B), and concentration of active constituent (C) on the strength of the product solution (measured in arbitrary units) in a recirculation unit.

42. 42 Example 15 Two blocks were used, with the ABC effect confounded with blocks, and there were two replications, resulting in the data in Figure 11.13. Data for Example 15 Figure 11.13 cont’d

43. 43 Example 15 The four block  replication totals are 288, 212, 88, and 220, with a grand total of 808, so cont’d

44. 44 Example 15 The other sums of squares are computed by Yates’s method using the eight experimental condition totals, resulting in the ANOVA table given as Table 11.14. By comparison with F.05,1,6 = 5.99, we conclude that only the main effects for A and C differ significantly from zero. Data for Example 15 Figure 11.14 cont’d

45. 45 Confounding Using More than Two Blocks

46. 46 Confounding Using More than Two Blocks In the case r = 2 (four blocks), three effects are confounded with blocks. The experimenter first chooses two defining effects to be confounded. For example, in a five factor experiment (A, B, C, D, and E), the two three-factor interactions BCD and CDE might be chosen for confounding. The third effect confounded is then the generalized interaction of the two, obtained by writing the two chosen effects side by side and then cancelling any letters common to both: (BCD)(CDE) = BE.

47. 47 Confounding Using More than Two Blocks Notice that if ABC and CDE are chosen for confounding, their generalized interaction is (ABC)(CDE) = ABDE, so that no main effects or two-factor interactions are confounded Once the two defining effects have been selected for confounding, one block consists of all treatment conditions having an even number of letters in common with both defining effects.

48. 48 Confounding Using More than Two Blocks The second block consists of all conditions having an even number of letters in common with the first defining contrast and an odd number of letters in common with the second contrast, and the third and fourth blocks consist of the “odd/even” and “odd/odd” contrasts.

49. 49 Confounding Using More than Two Blocks In a five-factor experiment with defining effects ABC and CDE, this results in the allocation to blocks as shown in Figure 11.14 (with the number of letters in common with each defining contrast appearing beside each experimental condition). Four blocks in a 2 5 factorial experiment with defining effects ABC and CDE Figure 11.14

50. 50 Confounding Using More than Two Blocks The block containing (1) is called the principal block. Once it has been constructed, a second block can be obtained by selecting any experimental condition not in the principal block and obtaining its generalized interaction with every condition in the principal block. The other blocks are then constructed in the same way by first selecting a condition not in a block already constructed and finding generalized interactions with the principal block.

51. 51 Confounding Using More than Two Blocks For experimental situations with p > 3, there is often no replication, so sums of squares associated with nonconfounded higher-order interactions are usually pooled to obtain an error sum of squares that can be used in the denominators of the various F statistics. All computations can again be carried out using Yates’s technique, with SSBl being the sum of sums of squares associated with confounded effects. When r > 2, one first selects r defining effects to be confounded with blocks, making sure that no one of the effects chosen is the generalized interaction of any other two selected.

52. 52 Confounding Using More than Two Blocks The additional 2 r – r – 1 effects confounded with the blocks are then the generalized interactions of all effects in the defining set (including not only generalized interactions of pairs of effects but also of sets of three, four, and so on).

53. 53 Fractional Replication

54. 54 Fractional Replication When the number of factors p is large, even a single replicate of a 2 p experiment can be expensive and time consuming. For example, one replicate of a 2 6 factorial experiment involves an observation for each of the 64 different experimental conditions. An appealing strategy in such situations is to make observations for only a fraction of the 2 p conditions. Provided that care is exercised in the choice of conditions to be observed, much information about factor effects can still be obtained.

55. 55 Fractional Replication Suppose we decide to include only 2 p – 1 (half) of the 2 p possible conditions in our experiment; this is usually called a half-replicate. The price paid for this economy is twofold. First, information about a single effect (determined by the 2 p – 1 conditions selected for observation) is completely lost to the experimenter in the sense that no reasonable estimate of the effect is possible.

56. 56 Fractional Replication Second, the remaining 2 p – 2 main effects and interactions are paired up so that any one effect in a particular pair is confounded with the other effect in the same pair. For example, one such pair may be {A, BCD}, so that separate estimates of the A main effect and BCD interaction are not possible. It is desirable, then, to select a half-replicate for which main effects and low-order interactions are paired off (confounded) only with higher-order interactions rather than with one another.

57. 57 Fractional Replication The first step in specifying a half-replicate is to select a defining effect as the nonestimable effect. Suppose that in a five-factor experiment, ABCDE is chosen as the defining effect. Now the 2 5 = 32 possible treatment conditions are divided into two groups with 16 conditions each, one group consisting of all conditions having an odd number of letters in common with ABCDE and the other containing an even number of letters in common with the defining contrast. Then either group of 16 conditions is used as the half-replicate.

58. 58 Fractional Replication The “odd” group is a, b, c, d, e, abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde, abcde Each main effect and interaction other than ABCDE is then confounded with (aliased with) its generalized interaction with ABCDE. Thus (AB)(ABCDE) = CDE, so the AB interaction and CDE interaction are confounded with each other.

59. 59 Fractional Replication The resulting alias pairs are {A, BCDE} {B, ACDE} {C, ABDE} {D, ABCE} {E, ABCD} {AB, CDE} {AC, BDE} {AD, BCE} {AE, BCD} {BC, ADE} {BD, ACE} {BE, ACD} {CD, ABE} {CE, ABD} {DE, ABC} Note in particular that every main effect is aliased with a four-factor interaction. Assuming these interactions to be negligible allows us to test for the presence of main effects.

60. 60 Fractional Replication To specify a quarter-replicate of a 2 p factorial experiment (2 p – 2 of the 2 p possible treatment conditions), two defining effects must be selected. These two and their generalized interaction become the nonestimable effects. Instead of alias pairs as in the half-replicate, each remaining effect is now confounded with three other effects, each being its generalized interaction with one of the three nonestimable effects.

61. 61 Example 16 The article “More on Planning Experiments to Increase Research Efficiency” (Industrial and Eng. Chemistry, 1970: 60–65) reports on the results of a quarter-replicate of a 2 5 experiment in which the five factors were A = condensation temperature, B = amount of material B, C = solvent volume, D = condensation time, E = amount of material E. The response variable was the yield of the chemical process. The chosen defining contrasts were ACE and BDE, with generalized interaction (ACE)(BDE) = ABCD.

62. 62 Example 16 The remaining 28 main effects and interactions can now be partitioned into seven groups of four effects each, such that the effects within a group cannot be assessed separately. For example, the generalized interactions of A with the nonestimable effects are (A)(ACE) = CE, (A)(BDE) = ABDE, and (A)(ABCD) = BCD, so one alias group is {A, CE, ABDE, BCD}. cont’d

63. 63 Example 16 The complete set of alias groups is {A, CE, ABDE, BCD} {B, ABCE, DE, ACD} {C, AE, BCDE, ABD} {D, ACDE, BE, ABC} {E, AC, BD, ABCDE} {AB, BCE, ADE, CD} {AD, CDE, ABE, BC} cont’d

64. 64 Fractional Replication Once the defining contrasts have been chosen for a quarter-replicate, they are used as in the discussion of confounding to divide the 2 p treatment conditions into four groups of 2 p – 2 conditions each. Then any one of the four groups is selected as the set of conditions for which data will be collected. Similar comments apply to a 1/2 r replicate of a 2 p factorial experiment.

65. 65 Fractional Replication Having made observations for the selected treatment combinations, a table of signs similar to Table 11.10 is constructed. Signs for Computing Effect Contrasts Table 11.10

66. 66 Fractional Replication The table contains a row only for each of the treatment combinations actually observed rather than the full 2 p rows, and there is a single column for each alias group (since each effect in the group would have the same set of signs for the treatment conditions selected for observation). The signs in each column indicate as usual how contrasts for the various sums of squares are computed. Yates’s method can also be used, but the rule for arranging observed conditions in standard order must be modified.

67. 67 Fractional Replication The difficult part of a fractional replication analysis typically involves deciding what to use for error sum of squares. Since there will usually be no replication (though one could observe, e.g., two replicates of a quarter-replicate), some effect sums of squares must be pooled to obtain an error sum of squares. In a half-replicate of a 2 8 experiment, for example, an alias structure can be chosen so that the eight main effects and 28 two-factor interactions are each confounded only with higher-order interactions and that there are an additional 27 alias groups involving only higher-order interactions.

68. 68 Fractional Replication Assuming the absence of higher-order interaction effects, the resulting 27 sums of squares can then be added to yield an error sum of squares, allowing 1 df tests for all main effects and two-factor interactions. However, in many cases tests for main effects can be obtained only by pooling some or all of the sums of squares associated with alias groups involving two-factor interactions, and the corresponding two-factor interactions cannot be investigated.

69. 69 Example 17 Example 16 continued… The set of treatment conditions chosen and resulting yields for the quarter-replicate of the 2 5 experiment were e ab ad bc cd ace bde abcde 23.2 15.516.9 16.2 23.8 23.4 16.8 18.1

70. 70 Example 17 The abbreviated table of signs is displayed in Table 11.15. Table of Signs for Example 17 Table 11.15 cont’d

71. 71 Example 17 With SSA denoting the sum of squares for effects in the alias group {A, CE, ABDE, BCD}, Similarly, SSB = 53.56, SSC = 10.35, SSD =.91 SSE =10.35 (the differentiates this quantity from error sum of squares SSE), SSAB = 6.66, and SSAD = 3.25, giving SST = 4.65 + 53.56 +…+ 3.25 = 89.73. To test for main effects, we use SSE = SSAB + SSAD = 9.91 with 2 df. cont’d

72. 72 Example 17 The ANOVA table is in Table 11.16. ANOVA Table for Example 17 Table 11.16 cont’d

73. 73 Example 17 Since F.05,1,2 = 18.51, none of the five main effects can be judged significant. Of course, with only 2 df for error, the test is not very powerful (i.e., it is quite likely to fail to detect the presence of effects). The article from Industrial and Engineering Chemistry from which the data came actually has an independent estimate of the standard error of the treatment effects based on prior experience, so it used a somewhat different analysis. cont’d

74. 74 Example 17 Our analysis was done here only for illustrative purposes, since one would ordinarily want many more than 2 df for error. cont’d